6 THE FREDHOLM INTEGRAL EQUATION 345

by (8.5.3). But the left-hand side is (U<p \ <p), and this violates the assumption
that U is a positive operator.

Remark now that for any finite number of eigenvalues A* (1 < k < ri)

n _

KW(J, 0 = KO, t) — j) 4 <Pk(f)9k(t) is the kernel function of a positive oper-

fc=i
ator £/„ , for we have

but the right-hand side of that equation can be written (Ug\g) with

n

g =/- £ (/| <pfc)<p*, as is readily verified, hence is positive by assumption.
jt=i

Therefore, by (5.3.1) it follows from Kn(s, s)^0 that the series £ An |<pB(j)|2

n
is convergent, and we have ]T !„ I^C?)!2 < K(j, j) for all sel. By Cauchy-

n

Schwarz, we conclude that

0

(11.6.7.1) E

for all Os, 0 e I x I. Hence, as K(^ t) is bounded in I, f or fixed s e I, the series
) *s uniformly convergent for /el. By (11.6.6), (8.7.8), and

(8.5.3), we conclude that £ ln <pn(s)(pn(t) = K(s, t) for all (s, t) e I x I since

n
2

t ->• K(5", 0 ~~ X ^" (Pn(s)(Pn(f) is continuous in I and its integral in I is 0. In

n

particular, we have K(s, s) = Ys^n\(Pn(s)\2l by Dini's theorem (7.2.2) the

n

series £^n|<pnCs)|2 l$ therefore uniformly convergent in I, and (11.6.7.1)

n ._____

proves that the series J] An (pn(s)(pn(t) is absolutely and uniformly convergent

n

in I x 1, which ends the proof.

jRtfWtfrfcs

(11.6.8) The result (11.6.7) is still true when we only suppose that t/has a
finite number of eigenvalues vk < 0 (1 < k < m). For (11.5.7(c)) shows then

that in the space F^+1, orthogonal supplement of E(v1) +-----h E(vm) in G,

the restriction of the operator U is positive, and we apply (11.6.7) to that <0t ian kernel, the series £ 1* is convergent and